A set of mobile robots (represented as points) is distributed in the Cartesian plane. The collection contains an unknown subset of byzantine robots which are indistinguishable from the reliable ones. The reliable robots need to gather, i.e., arrive to a configuration in which at the same time, all of them occupy the same point on the plane. The robots are equipped with GPS devices and at the beginning of the gathering process they communicate the Cartesian coordinates of their respective positions to a central authority. On the basis of this information, and without the knowledge of which robots are faulty, the central authority designs a trajectory for every robot. The central authority aims to provide the trajectories which result in the shortest possible gathering time of the reliable robots. The efficiency of a gathering strategy is measured by its competitive ratio, i.e., the maximal ratio between the time required for gathering achieved by the given trajectories and the optimal time required for gathering in the offline case, i.e., when the faulty robots are known to the central authority in advance. The role of the byzantine robots, controlled by the adversary, is to act so that the gathering is delayed and the resulting competitive ratio is maximized.

The objective of our paper is to propose efficient algorithms when the central authority is aware of an upper bound on the number of byzantine robots. We give optimal algorithms for collections of robots known to contain at most one faulty robot. When the proportion of byzantine robots is known to be less than one half or one third, we provide algorithms with small constant competitive ratios. We also propose algorithms with bounded competitive ratio in the case where the proportion of faulty robots is arbitrary.

一组移动机器人（表示为点）分布在笛卡尔平面中。该集合包含一个未知的拜占庭式机器人子集，与可靠的机器人无法区分。可靠的机器人需要聚集，即到达一个配置，在该配置中，所有机器人都同时占据飞机上的同一点。机器人配备了GPS设备，在收集过程开始时，它们会将各自位置的直角坐标传递给中央机构。根据此信息，并且在不知道哪个机器人有故障的情况下，中央机构会为每个机器人设计一个轨迹。中央机构旨在提供轨迹，从而使可靠机器人的采集时间最短。收集策略的效率由其竞争率来衡量，即竞争给定轨迹所需的收集时间与离线情况下收集所需的最佳时间（即已知故障机器人的最佳时间）之间的最大比率。中央预先。由对手控制的拜占庭式机器人的作用是行动，以使聚会被延迟，从而使竞争比最大化。

本文的目的是在中央主管部门意识到拜占庭式机器人数量的上限时提出有效的算法。对于已知最多包含一个故障机器人的机器人，我们给出了最佳算法。当已知拜占庭式机器人的比例小于一半或三分之一时，我们提供的算法具有较小的恒定竞争率。在故障机器人的比例是任意的情况下，我们还提出了具有有限竞争比的算法。